Sreelakshmi Manjunath

Stability and Performance Analysis of Compound TCP With REM and Drop-Tail Queue Management

We study Compound TCP (C-TCP), the default TCP in the Windows operating system, with Random Exponential Marking (REM) and the widely used Drop-Tail queue policy. The performance metrics we consider are stability of the queue size, queuing delay, link utilization, and packet loss. We analyze the following models: 1) a nonlinear model for C-TCP with Drop-Tail and small buffers; 2) a stochastic variant of REM along with C-TCP; and 3) the original REM proposal as a continuous-time nonlinear model with delayed feedback. We derive conditions to ensure local stability and show that variations in system parameters can induce a Hopf bifurcation, which would lead to the emergence of limit cycles. With Drop-Tail and small buffers, the Compound parameters and the buffer size both play a key role in ensuring stability. In the stochastic variant of REM, larger thresholds for marking/dropping packets can destabilize the system. With the original REM proposal, using Poincaré normal forms and the center manifold analysis, we also characterize the type of the Hopf bifurcation. This enables us to analytically verify the stability of the bifurcating limit cycles. Packet-level simulations corroborate some of the analysis. Some design guidelines to ensure stability and low latency are outlined.

FAST TCP: Some queueing models and stability

We analyse the stability of FAST TCP, a new proposal for TCP, in three limiting regimes. When the queue is modelled as an integrator, the fluid model may lose local stability as the feedback delay varies. Loss of local stability is shown to occur via a Hopf bifurcation, and we analytically characterise the asymptotic orbital stability of the bifurcating limit cycles. In an intermediate and a small buffer regime, larger thresholds for dropping packets can lead to a locally unstable regime. Packet-level simulations complement some of the analytical insights.

Analyses of compound TCP with Random Early Detection (RED) queue management

We study the performance of Compound TCP with Random Early Detection (RED) in three different limiting regimes. In the first regime, averaging over the queue size helps to decides the probability of dropping packets. Then, we consider a model where averaging over the queue size is not performed, but the queue is modelled as an integrator. Finally, we consider a model where the threshold for dropping packets is so small that it is not possible to model the queue as an integrator. In these three regimes, we derive sufficient, as well as necessary and sufficient conditions for local stability. These conditions help to capture the dependence of protocol and network parameters on system stability. We also show that in the event of loss of local stability, the Compound TCP-RED system undergoes a Hopf bifurcation which would lead to limit cycles. Some of the analytical results are corroborated using packet-level simulations.

FAST TCP: Some fluid models, stability and Hopf bifurcation ☆

Abstract We study FAST TCP, a new transmission control protocol that uses queueing delay as its feedback measure. We highlight two continuous-time models proposed for FAST that represent two operating regimes: (i) queueing delay forms a large component of the end-to-end delay, (ii) propagation delay is the dominant component of the end-to-end delay. These models when coupled with the integrator model for the queue, are shown to yield qualitatively similar results. We then study one of these models in different queueing regimes. In the scenario where the queue can be modelled as an integrator, we conduct a detailed local stability analysis. This yields strict bounds, on the system parameters and round-trip time, to ensure local stability. We show that the system undergoes a Hopf bifurcation, when these bounds are violated, leading to the emergence of limit cycles in the system dynamics. As limit cycles could be detrimental to network performance, we conduct a detailed Hopf bifurcation analysis using Poincare normal forms and center manifold theory. This enables us to characterise the type of the Hopf bifurcation and determine the orbital stability of the limit cycles. We then consider a regime with smaller queues, where end systems react primarily to packet loss. In this regime, larger thresholds could lead to instability. Packet-level simulations corroborate our analytical insights; non-linear oscillations are indeed observed in the queue size.

A proportionally fair controller with small Drop-Tail buffers: Local bifurcation analysis with two delays

We study a proportionally fair congestion controller with small Drop-Tail buffers. The model considers two sets of users, each with a different round-trip time, feeding into a single bottleneck link. Using a suitably motivated exogenous bifurcation parameter, we computationally exhibit the existence of a Hopf bifurcation. We then employ Poincaré normal forms and the center manifold theory, to provide an analytical basis to determine the type of the Hopf bifurcation. Using the analysis, one can also verify the orbital stability of the bifurcating limit cycles. Some numerical computations complement the theoretical analysis.

Performance analysis of compound TCP with a Proportional Integral (PI) control policy

In this paper, we study the performance of Compound TCP with the classical Proportional Integral (PI) control policy implemented at routers. We first conduct a local stability analysis and derive the necessary and sufficient condition for local stability of the non-linear model of Compound with PI. We explicitly show that the system undergoes a Hopf bifurcation as it transits into instability, which would lead to the emergence of limit cycles in the queue size. We then use Poincaré normal forms and center manifold theory to provide an analytical basis to characterise the Hopf bifurcation and determine the orbital stability of the limit cycles. The analysis is complemented with numerical examples.

Stability and Hopf bifurcation analysis of the Mackey-Glass and Lasota equations

Time delays are an integral part of various physiological processes. In this paper, we analyse two models for physiological systems: the Mackey-Glass and Lasota equations. We first exhibit a sufficient condition to ensure local stability, and then outline the associated necessary and sufficient condition for stability. Using a non-dimensional bifurcation parameter, we then highlight that stability will be lost via a Hopf bifurcation. We also explicitly characterise the type of the Hopf bifurcation using Poincaré normal forms and the center manifold theory. The theoretical analysis is complemented with some numerical examples, stability charts and bifurcation diagrams.

Local Hopf bifurcation analysis of logistic population dynamics models with two delays

Multiple time lags can occur very naturally in the study of population dynamics. In this paper, we study two forms of the delay logistic equation with two discrete time delays. For both the models, we identify the condition for the first local Hopf bifurcation. For our analysis, we employ a non-dimensional bifurcation parameter. Using Poincaré normal forms and the center manifold theory, we also conduct the requisite analysis to determine the type of the Hopf bifurcation. This enables us to determine the asymptotic orbital stability of the bifurcating periodic solutions. The analysis is complemented with some numerical examples and bifurcation diagrams.

Stability and bifurcation analysis of a Lotka-Volterra time delayed system

Models that describe interactions between species are often used to study the dynamics of populations in an ecosystem. In this paper we focus on one such model, i.e. a time delayed version of the Lotka-Volterra dynamical system. In particular, we study the effects of time delays in, (i) the interspecies interactions and, (ii) the carrying capacity of the prey population, on the system dynamics. In these two cases, we first perform a local stability analysis, where we derive the necessary and sufficient condition for stability. It is shown that, as the time delay is varied, the system loses stability in the first case and exhibits a finite number of stability switches in the second case. We then explicitly show that the loss of stability, in the first case, happens through a Hopf bifurcation. Further, using Poincaré normal forms and center manifold theorem, we analyse the type of the Hopf bifurcation. To complement our analysis, stability charts and bifurcation diagrams are also presented.

A Kaldor-Kalecki model of business cycles: Stability and limit cycles

Combining ideas proposed by Kaldor and Kalecki leads to a non-linear, time delayed, model for business cycle dynamics. In this paper, we analyse the stability and the local Hopf bifurcation properties of a Kaldor-Kalecki type model. In the analysis of such models, it is common to assume that the time delay continuously varies, and hence it is treated as a bifurcation parameter. However, this may not be a realistic assumption in various economic environments. We use an exogeneous, and non-dimensional, parameter as the bifurcation parameter to show that the underlying system undergoes a local Hopf bifurcation. Further, as this parameter gradually varies beyond the Hopf condition, we expect limit cycles to emerge from the stable equilibrium. We then, using Poincaré normal forms and the center manifold theory, outline the analysis to verify the type of the Hopf bifurcation and determine the stability of the limit cycles. The theoretical analysis is illustrated with some numerical examples.